**11. Conclusions**

270 Recent Advances in Crystallography

(1/4,1/4,0) given in Table 10.

1 0 0 0 0 1 0 0 0 0 1 0


2 [010]( 0 2 1) 1/2 \* 1/8,-1/8,1/4

½,1/2,1/2) (1/4,1/4,0)

2

<sup>1</sup>2 [-1-12]( 0 0 1) \*

6

conventional axes for the tetragonal space group.

1

5

group-theoretical approach is preferred. For this classification, each space group description is referred to a primitive base and an origin. Two space groups **G** and **G'** belong to the same space-group type if a transformation pair **P**, **p** exists, for which the 3x3 matrix has integral elements with det(**P**) = 1 and the **p** vector consists of three real numbers, such that **G** is transformed into **G**' (see, Wondratschek in ITA83). This definition is very simple, but

Let's modify the above equivalence definition for the practical purposes. Now **G'** means a conventionally described *space-group type*, represented by a unique set of generators or symmetry matrices. **G** is still referred to a primitive base. **G** belongs to the space-group type **G'** if a transformation pair **P**, **p** exists, for which the 3x3 matrix has integral elements with det(**P**) = 1,2,3 or 4 and the **p** vector consists of three real numbers, such that **G** is transformed into **G**'. The first step in determining the type of group **G** is to refer it to a centred Bravais base by a proper selection of coordinate axes. It is simple with the help of dual symbols what will be illustrated by the group description TSG = *I*4122 (1,0,0; 0,1,0; ½,1/2,1/2)

3

7

**Table 10.** Symmetry matrices and dual symbols of the group description TSG = *I*4122 (1,0,0; 0,1,0;

Items 1,2,3 and 4 define symmetry axis 41 parallel to [1�1�2]. Matrices (5,6) and (7,8) describe two pairs of orthogonally oriented twofold axes, according to property *u*1*h*2 + *v*1*k*2 + *w*1*l*2 = *u*2*h*<sup>1</sup> + *v*2*k*1 + *w*2*l*1 = 0 between the corresponding splitting indices. A similar test shows the orthogonality between 41 and all twofold axes. Thus, two orthogonal bases and two transformation matrices may be constructed from [*uvw*] indices. The first transformation matrix **P** with columns [100], [010], [1�1�2] has det(**P**) = 2 and leads to *I*-centred basis. The equivalent F-centred Bravais tetragonal cell is involved with P in the form [11�0], [110] and [1�1�2]. The application of the first transformation matrix is equivalent to the selection of the

Moreover, the arithmetic type 422*I* of the analysed group is determined. From the predefined data one can find that only two space-group types, namely the groups with sequence numbers 97 and 98, belong to this arithmetic type. Group 97 is symmorphic. Thus,

0 -1 -1 0,25 1 0 0 0,25 0 0 1 0,5

4+[-1-12]( 0 0 1) 1/4 \* 0,1/2,0

0 1 1 0 1 0 1 0 0 0 -1 0

1/8,3/8,1/4 2 [110]( 1 1 1) 2 [1-10] \* 1/4,1/4,0

4

8

0 1 0 0,75 -1 0 -1 0,75 0 0 1 0,5

4-[-1-12]( 0 0 1) 1/4 \* 1,0,0

0 -1 0 0,5 -1 0 0 0,5 0 0 -1 0

finding the transformation between two sets of matrices may be a real challenge.


1/4,1/4,0

1 0 1 0,75 0 -1 0 0,75 0 0 -1 0,5

2 [100]( 2 0 1) 1 \* -

It was assumed that, the choice of the known or modified algorithms should be motivated by obtaining functional relations f1 (*SG symbol*, *ordering number*) = *symmetry matri*x and f2 (*symmetry matrix*) = *geometric description* on the assumption that the needed conventions are reduced to a minimum. This unique arithmetic and geometric description of space groups cannot be obtained at the cost of excessive amount of the predefined data, ineffective or sophisticated algorithms. Moreover, in the case of conventional axes and origins the results cannot differ from that contained in ITA83. It appears that such practical purposes have been achieved.

The commonly accepted Hermann-Mauguin symbol is very informative and useful for controlling group orientation, but is not dependent on the origin choice and is not applicable for a non-conventional space group description. The absolute position of symmetry elements needs an explicit origin specification, by giving a complete information about the translation part of generators. The coding of such generators leads to the elaborated symbols (Zachariassen/Shmueli) or to the concise symbols (Hall), but involved with many conventions and rather sophisticated interpretation of symbols. Moreover, in universal approaches based on arbitrary generators most of the computing time is spent on tests for closure or for redundancy of generated operations.

The '*transformational concept*' presented in this chapter introduces an 'absolute' description of each space group type contrary to the multiple standards for some space group types in ITA83. Thus, all groups belonging to a given SG type may be derived from the same set of optimally selected generators, assuming that the transformation from the SG type to a needed description is known. The TSG symbol was aimed at pointing in a computer program at the set of generators, prepared as 'composition series', and at giving an explicit definition of necessary transformation of these generators. There is no need to interpret the TSG symbol or tests the generators. If generators published in ITA83 serve as predefined type generators, nearly 100% of computing time is spent on non-redundant SG derivation. Programs based on a transformational concept reproduce all conventional descriptions given in ITA83; in the non-conventional cases they lead to descriptions in the ITA83 style.

#### 272 Recent Advances in Crystallography

From this point of view, such algorithms standardize any SG descriptions and their geometric interpretations and may be treated as an electronic extension of the printed ITA83 tables. In this context the classification of space groups into conventionally versus nonconventionally described, and the evolutionary changes between different editions of Tables are unsubstantial.

The generation schema based on composition series leads also to a considerable reduction of predefined data, what is an important feature in the implementing and testing of SG derivation programs. Since all the space groups which are based on a given point group are generated in the same way, only the translational parts of generators should be stored. Using small improvements discussed in the text, the non-redundant number of composition series generators is 3, and thus the non-redundant number of translations for each spacegroup type is 0,1 or maximum 2.

The introduced dual symbol of symmetry operation sheds some light on the lattice properties and thus changes a rather informative character of the geometric symbols into a valuable tool for finding the transformation between different descriptions of the same groups. Since the symbol is universal, convention free, easy to derive and to manipulate on a computer, it is advised to generate space group descriptions simultaneously in two forms, as coordinate triplets and as dual symbols. Any new information can be obtained from the additional Seitz symbol.

The simplicity of the considered approach is evident and seems to be valuable also for teaching purposes. Algorithms were implemented in the Visual Basic, in the Excel environment. All procedures need less than 600 programming lines including 200-line subroutine for transforming a symmetry matrix into a dual symbol. This procedure may be also used as stand-alone item to characterize any symmetry matrix. The source code included in the Excel worksheet may be obtained from the author on request.
